Lecture 13 More on hypothesis testing

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1 Lecture 13 More on hypothesis testing Thais Paiva STA Summer 2013 Term II July 22, / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

2 Lecture Plan 1 Type I and type II error 2 Hypothesis test for a proportion 2 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

3 Hypothesis testing: ingredients 1 Null Hypothesis (H 0 ): Hypothesis that is falsifiable by data 2 Alternative Hypothesis (H A ): Hypothesis to be accepted if H 0 is rejected 3 Significance Level (α): The probability of rejecting H 0 when H 0 is true 4 Test statistic: A statistic derived from the sample to determine whether H 0 should be rejected 5 P-value: The probability of observing a test statistic as extreme (or more extreme) as the observed test statistics if H 0 were true 3 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

4 Hypothesis testing: steps First of all, write the null and alternative hypotheses. H 0 : µ = µ 0 H A : µ > µ 0 P-value: 1 Calculate the test statistic T = X µ 0 s/ n 2 Find P(t n 1 > T ), the p-value 3 Reject H 0 if the p-value α Critical value: 1 Calculate the test statistic T = X µ 0 s/ n 2 Use α to find the critical value t such that P(t n 1 > t ) = α 3 Reject H 0 if T t 4 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

5 Example: age at first child Based on the data from the National Vital Statistics Report, Vol. 48, No. 14 (1999), it is claimed that the average age of a woman at her first pregnancy is greater than the 1990 mean age of 24.6 years. A random sample of 40 women who gave birth to their first child in 1999 is analyzed: The sample mean age is 27.1 years The sample standard deviation is 6.4 years Test the above claim at the α = 0.05 level. 5 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

6 Example: age at first child H 0 : µ = 24.6 H A : µ > 24.6 T = X µ 0 s n = = 2.47 t 1 α,39 = t 0.95,39 = T > t 1 α, > Reject H 0 or equivalently P[t 39 > T ] = < 0.05 Reject H 0 6 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

7 Hypothesis Testing and Error Remember, when we perform a test, we have a null hypothesis H 0 and an alternative hypothesis H A. H 0 is either true or false. Let s consider the case that H 0 is true. Reject H 0 Fail to reject H 0 H 0 is true Type I error 7 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

8 Hypothesis Testing and Error Type I Error: α False rejection Reject the null hypothesis when it is true This is always pre-defined before an experiment or an analysis Typically, α = 0.10, 0.05, or The choice depends on how often I am willing to tolerate rejecting the null hypothesis incorrectly Determines statistical significance level (1 α) = probability of not rejecting the null 8 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

9 Hypothesis Testing and Error Now let s consider the case that H 0 is false. Reject H 0 Fail to reject H 0 H 0 is true Type I error H 0 is false Type II error This is the table you should memorize. 9 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

10 Hypothesis Testing and Error Type II Error: β False acceptance (really failing to reject) Do not reject the null hypothesis when it is false We cannot set this error in advance! It depends on what the true value of the unknown parameter, e.g. µ, is (1 β) = probability of true rejection. This is also known as the power of a test We want to have high power 10 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

11 Interpreting α and β 11 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

12 How does it work? 12 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

13 Reducing α? 13 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

14 Hypothesis Testing and Error Consider a population with mean µ and σ = 2. We draw a sample of size 10 and assume we know the true population variance. We now perform a hypothesis test at significance level 0.10 with Then H 0 : µ = 0 H A : µ > 0 The critical value is Z = 1.28 (from Table IV in the book Normal table) The test statistic is Z test = Under the null, X 0 2/ = X X 10 2 N(0, 1) 14 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

15 Hypothesis Testing and Error But if the true mean is, say, µ = 0.5, the sampling distribution of X is X 0.5 2/ N(0, 1) 10 X 0.5 2/ 10 = X 10 2 ( X 10 2 N ) , 1 2 which was our test statistic. Remember, under the null, X 10 2 N(0, 1), 15 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

16 Hypothesis Testing and Error Because in reality µ = 0.5, Z test = X 10 2 N ( 0.5 ) 10, 1 2 The power is probability of rejecting the null when the null is false. To reject the null, I need to observe a test statistic higher than the cutoff under the null. ( ) X 10 P(Z test > Z ) = P > 1.28 Not much power at all. = P 2 ( X = P(Z > 0.489) = ) > / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

17 Hypothesis Testing and Error Let s repeat the calculation replacing our fictitious µ = 0.5 with some µ = µ true, σ = 2 with some σ, and α = 0.1 with any α. Z test = X ( ) n µ true n N, 1 σ σ Then, the power is ( ) X n P(Z test > Z ) = P > Z α σ ( X n µtrue n = P > Z α µ ) true n σ σ ( = P Z > Z α µ ) true n σ 17 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

18 Hypothesis Testing and Error Again, the power is ( P Z > Z α µ ) true n σ Notice that the power increases when The sample size n and the true mean µ increase The population variance σ 2 and Z α decrease The type I error rate α increases 18 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

19 Power and sample size calculation Again, the power is ( Power = P Z > Z α µ ) true n σ Here s the key: If I know α, σ, and µ true, I can determine what sample size I need to achieve a desired power Clearly, I do not know these quantities, but it is usually useful to guess 19 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

20 Sample size calculation: example Let s say we work for a drug company, and we think we have made an effective cholesterol drug. We call it effective if the drug has a significant effect that is, H 0 : µ = 0 H A : µ > 0 However, it is very expensive to run clinical trials on a new drug, so we really do not want too many false discoveries. We insist on a power of 98%. 20 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

21 Sample size calculation: example If we expect the effect to be about 5 points on the cholesterol scale (µ true ) with a standard deviation of 7, and we want a 0.05 significance level, how many patients do we need to use? ( Power = P Z > Z α µ ) true n σ From the table: 0.98 = P ( Z > Z n 7 ) so we find that n = = n 7 n = / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

22 Testing for a single proportion Remember, a proportion is just a mean in disguise. Let X 1,..., X n be iid Bernoulli(p). Then ˆp = 1 n n i=1 X i Let s test H 0 : p = p 0 for some p 0 against H a : p > p 0 22 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

23 Testing for a single proportion By the CLT, One test statistic would be ( ˆp N p, ) p(1 p) n T = ˆp p 0 ˆp(1 ˆp) n This is correct, but we can do better. t n 1 23 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

24 Testing for a single proportion Because both the unknown mean and variance are a function of the quantity we are testing, p, we know the variance under the null hypothesis. It is p 0 (1 p 0 ). So, we can use the normal distribution instead of the t to get better power: under the null. Now, all we need to do is Z = ˆp p 0 p 0(1 p 0) n N(0, 1) Look up a p-value corresponding to Z or Pick α, find a critical value Z, and compare it to Z 24 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

25 Testing for a single proportion: example Market researchers want to know if shoppers are sensitive to the price of their groceries. They collected a sample of 802 shoppers, and found that 378 of them could correctly name the price of the items in their carts. They want to know if at least half of the shoppers are sensitive to prices: H 0 : p 0.5 H A : p > / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

26 Testing for a single proportion: example We compute the test statistic using ˆp = n = 802 Then Z = (1 0.5)/ , giving a p-value of Does this make sense? 26 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

27 Summary 1 There are two kinds of errors we can make 2 We set the type I error rate beforehand to a level we are willing to accept 3 The type II error rate (and hence the power) depends on things we do not know 4 Hypothesis tests for a proportion are basically the same as hypothesis tests for a mean 27 / 27 Thais Paiva STA Summer 2013 Term II Lecture 13, 07/22/2013

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